Action Noether Time Translation Invariance Implies Energy Conservation

A theorem about time symmetry

In classical mechanics, the action of a trajectory is a single number that summarizes the whole path, usually the integral of kinetic minus potential energy over time. Emmy Noether proved in 1918 that every continuous symmetry of the action corresponds to a conserved quantity. Time-translation invariance, meaning the action is unchanged if you start the same motion later or earlier, yields conservation of energy. This is one of the most used results in physics, and it applies to any system whose laws do not change with time.

The Recognition Science framework formalizes this result in its machine-checked library of formal theorems. The declaration time_translation_invariance_implies_energy_conservation states that if a real-valued action functional is invariant under the time-translation flow, then the action itself is conserved along that flow. The proof is a direct corollary of a more abstract Noether theorem already in the library, and the declaration carries no axioms beyond the standard ones. In plain terms: if the rules of the game do not depend on when you play, the total energy of the motion stays constant.

The framework also specializes this to the standard mechanical Lagrangian L = ½ m q̇² − V(q). When the potential V does not depend explicitly on time, the theorem energy_conservation_of_J_action proves that the total energy E = T + V is the same at any two times along a trajectory that satisfies the Euler-Lagrange equation. This is the familiar statement that energy is conserved in a time-independent potential, now checked in the framework's library.

What the declaration does not claim is just as important. It does not prove that every physical system has time-translation invariance; that is a separate assumption. It does not define what "energy" means for arbitrary actions; the interpretation as total energy is attached to the standard mechanical case. And it does not extend to discrete time steps or to systems where the action depends on time explicitly. The theorem is a precise conditional: if the symmetry holds, then the conservation follows.

/-- **Energy conservation from time-translation invariance.**

    If a J-action functional is time-translation invariant, then by
    `noether_core` it is itself conserved along the time-translation flow.
    The conserved quantity is interpreted as the total energy. -/
theorem time_translation_invariance_implies_energy_conservation
    (S : RealAction → ℝ)
    (h_inv : ∀ t, IsSymmetryOf (timeTranslationFlow.flow t) S) :
    IsConservedAlong S timeTranslationFlow.flow :=
  noether_core h_inv

/-- **Energy conservation from time-translation invariance.**

    If a J-action functional is time-translation invariant, then by
    `noether_core` it is itself conserved along the time-translation flow.
    The conserved quantity is interpreted as the total energy. -/
theorem time_translation_invariance_implies_energy_conservation
    (S : RealAction → ℝ)
    (h_inv : ∀ t, IsSymmetryOf (timeTranslationFlow.flow t) S) :
    IsConservedAlong S timeTranslationFlow.flow :=
  noether_core h_inv

/-- **The standard total energy of mechanical motion is conserved when
    the potential is time-independent.**

    This is the concrete Noether theorem for the standard mechanics
    Lagrangian `L = ½ m q̇² - V(q)`: time-translation invariance is
    automatic when `V` does not depend on `t` explicitly, and energy
    conservation `E = T + V` follows.

    Proven directly by `Action.Hamiltonian.energy_conservation`, this
    lemma packages the result in the `Noether` namespace for clarity. -/
theorem energy_conservation_of_J_action (m : ℝ) (hm : 0 < m) (V : ℝ → ℝ)
    (γ : ℝ → ℝ)
    (hV_diff : ∀ t, DifferentiableAt ℝ V (γ t))
    (hγ_diff : ∀ t, DifferentiableAt ℝ γ t)
    (hγ_diff2 : ∀ t, DifferentiableAt ℝ (deriv γ) t)
    (h_dE_eq_factored : ∀ t : ℝ,
      deriv (HamiltonianMech.totalEnergy m V γ) t =
        deriv γ t * (m * deriv (deriv γ) t + deriv V (γ t)))
    (hEL : ∀ t : ℝ, QuadraticLimit.standardEL m V γ t = 0) :
    ∀ t₁ t₂ : ℝ,
      HamiltonianMech.totalEnergy m V γ t₁ = HamiltonianMech.totalEnergy m V γ t₂ :=
  HamiltonianMech.energy_conservation m hm V γ hV_diff hγ_diff hγ_diff2 h_dE_eq_factored hEL

What this page does not claim

The theorem does not prove that every physical system is time-translation invariant. The declaration does not define energy for arbitrary actions; the energy interpretation is specific to the standard mechanical Lagrangian. The theorem does not apply to systems where the action depends explicitly on time.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Action/Noether.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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